It's hard to prove that there isn't a way to do something, but I think the answer is no.
The saturation conjecture, now a theorem of Knutson and Tao, says that $c_{(N \alpha) (N \beta)}^{N \gamma} >0$ implies $c_{\alpha \beta}^{\gamma} >0$ for any positive integer $N$ and any partitions $\alpha$, $\beta$ and $\gamma$. Note that the corresponding statement for eigenvalues of Hermitian matrices is obvious. I suspect that any simple answer to your question would lead to a simple proof of the saturation conjecture and, while several proofs are now known, I would describe none of them as simple.
There is a relationship in the other direction. If $\alpha$, $\beta$ and $\gamma$ are partitions with $d$ parts then $c_{(N \alpha) (N \beta)}^{N \gamma}$ is a polynomial of degree $\binom{d-1}{2}$ in $N$ and the leading term (up to constants I'm not going to remember) is the volume of the space of triples $(A,B,C)$ of Hermitian matrices with spectra $(\alpha, \beta, \gamma)$ and $A+B=C$. So computing $c_{(N \alpha) (N \beta)}^{N \gamma}$ for enough values of $N$ allows you to compute the volume of a space of Hermitian matrices.
As a heurisitic, Hermitian matrix questions are about computing volumes of regions in $\mathbb{R}^M$; LR questions are about counting lattice points in those regions. Either one approximates the other, and the asymptotic behavior of lattice point counts tells you about volumes, but just knowing volumes will never tell you lattice point counts.
My favorite survey on the relation between the two problems is this one.
